Find the compound interest on Rs 16,000 invested at 20% per annum for 9 months, if the interest is compounded quarterly (every 3 months).

Introduction / Context:
This question involves compound interest over a fractional time period, where the nominal annual rate is 20% and compounding is quarterly. The total duration is 9 months, which corresponds to three quarters of a year. Because the compounding frequency matches the fraction of the year, we can treat the duration as an integer number of compounding periods, which simplifies the calculation. We must compute the final amount after 3 quarters and then subtract the principal to obtain the compound interest.

Given Data / Assumptions:

• Principal P = Rs 16,000.
• Nominal rate r = 20% per annum.
• Compounding frequency: quarterly.
• Total time = 9 months = 3 quarters.
• We need to find the compound interest (CI) for this period.

Concept / Approach:
For quarterly compounding, the period rate is r/4 per quarter, and the number of compounding periods is 4 * (time in years). Here, r/4 = 20/4 = 5% per quarter. The time is 9 months, which is 9/12 = 3/4 years, corresponding to 3 quarters. Thus, we use A = P * (1 + 0.05)^3. Once we find the amount A, the compound interest is CI = A - P. The numbers are manageable, and we can compute (1.05)^3 exactly with careful multiplication.

Step-by-Step Solution:
Periodic rate per quarter = 20% / 4 = 5% = 0.05. Number of quarters in 9 months = 3. Amount A after 3 quarters = 16,000 * (1.05)^3. Compute (1.05)^2 = 1.1025. Now (1.05)^3 = 1.1025 * 1.05 = 1.157625. Hence A = 16,000 * 1.157625 = 18,522 (approximately). Compound interest CI = A - P = 18,522 - 16,000 = 2,522. Therefore, the CI for 9 months is Rs 2,522.

Verification / Alternative check:
We can also compute stepwise for each quarter. After the 1st quarter: 16,000 * 1.05 = 16,800. After the 2nd quarter: 16,800 * 1.05 = 17,640. After the 3rd quarter: 17,640 * 1.05 = 18,522. This gives the same final amount. Subtracting Rs 16,000 again yields Rs 2,522 in interest. Since both the direct power method and stepwise method agree, the result is reliable.

Why Other Options Are Wrong:
Rs 2,422 and Rs 2,322 are slightly smaller and would correspond to a lower effective factor than (1.05)^3, implying a rate below 20% nominal. Rs 3,522 and Rs 2,800 are larger than the correctly computed interest and would result from incorrect rates or more than three compounding periods. Hence, only Rs 2,522 matches the exact exponential growth for 3 quarters at 5% per quarter.

Common Pitfalls:
Some test-takers mistakenly treat 9 months as 0.75 years and then apply simple interest or annual compounding rather than quarterly compounding. Another common error is to miscount the number of quarters or to use 20% as the quarterly rate rather than 5%. Always convert months into the correct number of compounding periods and divide the annual nominal rate by the number of periods per year before using the compound interest formula.

Final Answer:
The compound interest earned on Rs 16,000 for 9 months is Rs 2,522.